Integrals: Numerical Problems Set (11)

Question

Evaluate the definite integral:

$\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} \, dx$

Solution — Step by Step

Why This Works

The “king property” exploits the symmetry of the interval $[0, a]$ about its midpoint. Many integrals that look impossible to evaluate by direct anti-differentiation fall instantly to this trick, especially trig integrals on $[0, \pi/2]$.

The pattern to recognise: an integrand of the form $\frac{f(x)}{f(x) + g(x)}$ where $f$ and $g$ swap under the substitution $x \to a - x$. Adding the two forms makes the denominator cancel.

Alternative Method

You could attempt direct anti-differentiation: write $\sin x = \frac{1}{2}[(\sin x + \cos x) + (\sin x - \cos x)]$, split the fraction, and integrate. The $\sin x - \cos x$ piece integrates to a logarithm. It works, but takes 5x longer than the king property.

Common Mistake

Students sometimes forget that the king property requires the limits to be $[0, a]$ (or, equivalently, $[a, b]$ with the substitution $x \to a + b - x$). Applying it to $[1, 3]$ without adjusting the substitution gives wrong answers.